Examples
The following section provides various examples for the usage of
km.surv.
Example 1
Qin et al.3 derived the probability mass function of
the KMPLE for one particular setting where there are n = 3
items on test, the failure times \(T_1,T_2\) and \(T_3\) and the censoring times \(C_1,C_2\) and \(C_3\) both follow an exponential(1)
distribution. The fixed time of interest is \(t_0 = -\ln(1/2)/2\), which is the median of
\(X = \min(T, C)\), where \(T\) is the failure time and \(C\) is the censoring time under a
random-censoring scheme. Therefore, perc = 0.5.
In this case, since failure and censoring times have the same
exponential distribution, they are equally likely to occur; that is,
h = 1/2.
For this example, km.surv is called with the arguments
n = 3, h = 1/2. To compare this with Example 1
in the km.pmf vignette, look at the plot where the cumulative
probability of X = 0.5 on the \(x\)-axis. Since the default of
lambda = 10, the times of interest are 0 to 1 at every 10th
percentile.
library(conf)
# display the probability mass functions at various times of interest
km.surv(n = 3, h = 1/2)
Example 2
A more interesting example is with n = 4 and two
probabilities of failure. For the first plot set a probability of
failure h = 1/3. Increasing lambda to 100 and
including the expected values connected by red lines produces a very
interesting plot. The probability mass functions have larger
probabilities of 1 due to the higher rate of censoring. The KMPLE
remains at 1 until the first failure so all possible censored items that
come before that first failure is considered in this probability. The
high probability of right-censored items is also evident at the end of
the experiment when there is a high probability that the last item is
censored resulting in a high probability that there will be an NA.
# display the probability mass function at various times of interest
# with the expected values in red connected with lines
km.surv(n = 4, h = 1/3, lambda = 100, ev = TRUE, line = TRUE)
title("High Censoring Rate")
In contrast with the high probability of right-censoring, the high
probability of a failure h = 2/3 results in the following
plot. We see an initial high probability of 1 that decays quicker since
there is less chance of there being censored items before the first
failure and a low probability of NA at the end of the experiment since
there is a higher probability that the last item will fail over being
censored.
# display the probability mass function at various times of interest
# with the expected values in red connected with lines
km.surv(n = 4, h = 2/3, lambda = 100, ev = TRUE, line = TRUE)
title("High Failure Rate")
Example 3
The function km.surv provides many arguments to make the
plot as useful as possible. For example, when n is larger,
the plot may be improved by using decimals instead of the exact
fractions (xfrac = FALSE) or gray dots where the intensity
is related to the probability instead of the size
(graydots = TRUE). When probabilities are too small to be
seen, gray outlines circle them. This option can be turned off with
gray.outline = FALSE. The size of the dots can be made
smaller or larger using gray.cex where the default is
1.
# display the probability mass function at various times of interest
km.surv(n = 7, h = 3/4, lambda = 50, graydots = TRUE, xfrac = FALSE)
Removing the outlines that accentuates the small probabilities produces a less busy plot.
# display the probability mass function at various times of interest
km.surv(n = 7, h = 3/4, lambda = 50, graydots = TRUE, xfrac = FALSE, gray.outline = FALSE)
Removing the outlines, increasing the dot size, and adding expected values to a plot with sample size of 5 and a slighter higher rate of failure than censoring, produces the following plot.
# display the probability mass function at various times of interest
km.surv(n = 5, h = 5/8, lambda = 30, graydots = TRUE, ev = TRUE, gray.outline = FALSE, gray.cex = 1.25)